The Numerical Scheme Satisfying Multiple Conservation Laws For Two Dimensional Linear Advection Equations

In this paper,we design a numerical scheme satisfying two conservation laws for linear advection equation.The numerical solutions of the scheme is better than that of traditional schemes at both accuracy and long-time numerical simulation.The first work of this paper is the theoretical analysis of the step-reconstruction scheme satisfying two conservation laws for one-dimensional linear advection equation,and prove that the schemes show the superconvergence property.It can be seen that the numerical errors of the step-reconstruction scheme produced in different time steps are not simply linear accumulated,but can cancel each other.It is this feature of self-canceling which makes the numerical solutions of the step-reconstruction scheme have the second order accuracy in the region far away from the extreme point.The theoretical conclusion is consistent with the numerical results.The second work of this paper is to develop the step-reconstruction scheme which satisfying two conservation laws to the two-dimensional linear advection equation.We adopt the Godunov splitting strategy to split the two-dimensional linear advection equation into two conserved one-dimensional linear advection equations,and design a step-reconstruction scheme which satisfying two conservation laws for the two conserved one-dimensional equations.The scheme calculates two numerical entities which are the numerical solution and the numerical energy.The numerical results show that the numerical scheme also have the property of super-convergence and can keep the structure of the solution well in numerical simulations.